Question

Radioactive isotope carbon decays at a rate proportional to the amount present. If the decay rate is 10% per thousand years and the current mass is 150mg what will the mass be 3 thousand years from now?

Answer 1

Carbon-14 dating is an important method of radioactive dating. It is based on the decay of carbon-14 nuclei, which have a half-life of 5730 years. When an organism dies, carbon exchange with the environment ceases, and the ¹4C nuclei in the organism decay as well.

Step-by-step explanation:

The decay of a radioactive isotope, such as carbon, follows an exponential decay model. The decay rate of 10% per thousand years means that every thousand years, the amount of the substance decreases by 10%.

The formula for exponential decay can be expressed as:

A(t)=A0⋅e^−kt

Where:

A(t) is the amount of substance at time

A0 is the initial amount of substance

k is the decay constant

e is the base of the natural logarithm (approximately 2.71828)

t is time

Given:

Decay rate = 10% = 0.10

Time = 1000 years

k= 1000ln(1−0.10)

k= 1000ln(0.90)

k≈−0.0001054

Now, to find the mass of the isotope 3,000 years from now:

Given current mass

A0=150 mg

A(3000)≈109.32mg

Therefore, the mass of the radioactive isotope carbon 3,000 years from now will be approximately 109.32 mg.